#230079

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×AGL(1,ℤ/8)	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,10,5,9,4,8)(3,7,11,14,12,15,13,16)(17,18,23,28,24,20,22,27)(19,29,31,25,32,21,26,30);(1,2,4,8,5,9,6,10)(3,7,13,16,12,15,11,14)(17,19)(18,25)(20,30)(21,27)(22,31)(23,26)(24,32)(28,29);(1,3)(2,7)(4,13)(5,12)(6,11)(8,16)(9,15)(10,14)(17,20)(18,24)(19,30)(21,26)(22,28)(23,27)(25,32)(29,31);(1,4,5,6)(2,8,9,10)(3,11,12,13)(7,14,15,16)(17,21,23,30,24,29,22,25)(18,26,28,19,20,31,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dei,e^b=eh,g^b=gh,d^c=di,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137275);